System S1 (Lewis)

Notes

James Dugundji
[Dugundji,
1940, p150-151]
proved that there is no finite Characteristic Matrix for S1.

M. Wajsberg showed ("Ein erweiterter Klassenkalkül" in
Monatshefte für Mathematik und Physik, vol 40 (1933), p 118f) that
if a>b is a (PC) tautology, then a=>b is a
theorem in S3 (not S1).
[Simons,
1953, p310].
But Simons goes on to say that Parry extended this to S1.
[Parry,
1939, p143, last paragraph,
and the footnote.] (The paragraph says that his proof is only valid for
S3 and above, but the footnote says how to prove it
for S1 and above from Wajsberg's original paper.

Note that the original also had: ~ ~ p => p, but that was shown
redundant by J. C. C. Mckinsey, "A reduction in the number of postulates
for C. I. Lewis' system of strict implication" in the "Bulletin of
the American Mathematical Society",
Vol. 40 (1934) pgs 110-112

[Lewis and Langford,
1932, p500]
Systems S1 through S5 are defined in the last few pages
of the last appendex of the book (p500-502).
And not directly by a list of axioms of each systems, but by
prose that says things like "System S1 is axioms B1-B7".
The common axiom list is on page 493.
Adjunction, Modus Ponens (Called the "rule of inference"),
and "substitution of equivalents" are on pg. 494.
Uniform substitution is implicitly included, as introduced on page 125.
The definitions above are introduced in Chapter 11, P123-124
(As syntatic equalities, but the '=' sign was stated (p123) to mean logical
equivalences.)